In the second chapter, we initiate the study of computable presentations of real and complex C-star-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C-star-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C-star-algebras. On the other hand, we show every finite-dimensional C-star-algebra is computably categorical.

In the third chapter, building on previous work by Franklin et al., we show if X is a compact subset of R which consists of the Cantor set and only finitely many intervals and points, then C(X; R) is arithmetically categorical as a real Banach space and C(X; C) is arithmetically categorical as a complex C-star-algebra. For any compact subset X ⊆ R, we show there is an arithmetical embedding of X into R in computable presentations of C(X; R). As a consequence, if C(X; R) or C(X; C) admit a computable presentation, then X is homeomorphic to an arithmetically closed subset of R. Also, multiplication is arithmetical in any computable presentation of C(X; R).